We develop some aspects of the theory of D-modules on schemes and indschemes of profinite type. These notions are used to define D-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. We also extend the Fourier-Deligne transform to Tate vector spaces. Let N be the maximal unipotent subgroup of a reductive group G. For a non-degenerate character χ : N ((t)) → G a , and a category C acted upon by N ((t)), there are two possible notions of the category of (N ((t)), χ)-objects: the invariant category C N ((t)),χ and the coinvariant category C N ((t)),χ . These are the Whittaker categories of C, which are in general not equivalent. However, there is always a family of functors T k : C N ((t)),χ → C N ((t)),χ , parametrized by k ∈ N. We conjecture that each T k is an equivalence, provided that the N ((t))-action on C is the restriction of a G((t))-action. We prove this conjecture for G = GL n and show that the Whittaker categories can be obtained by taking invariants of C with respect to a very explicit pro-unipotent group subscheme (not indscheme) of G((t)).
The notion of Hochschild cochains induces an assignment from Aff, affine DG schemes, to monoidal DG categories. We show that this assignment extends, under appropriate finiteness conditions, to a functor H : Aff → Alg bimod (DGCat), where the latter denotes the category of monoidal DG categories and bimodules. Any functor A : Aff → Alg bimod (DGCat) gives rise, by taking modules, to a theory of sheaves of categories ShvCat A .In this paper, we study ShvCat H . Vaguely speaking, this theory categorifies the theory of D-modules, in the same way as Gaitsgory's original ShvCat categorifies the theory of quasi-coherent sheaves. We develop the functoriality of ShvCat H , its descent properties and the notion of H-affineness.We then prove the H-affineness of algebraic stacks: for Y a stack satisfying some mild conditions, the ∞-category ShvCat H (Y) is equivalent to the ∞-category of modules for H(Y), the monoidal DG category of higher differential operators. The main consequence, for Y quasi-smooth, is the following: if C is a DG category acted on by H(Y), then C admits a theory of singular support in Sing(Y), where Sing(Y) is the space of singularities of Y.As an application to the geometric Langlands program, we indicate how derived Satake yields an action of H(LSǦ) on D(Bun G ), thereby equipping objects of D(Bun G ) with singular support in Sing(LSǦ). 1.1.2. Tannaka duality. For Y an algebraic stack satisfying mild conditions, Tannaka duality ([Lur18, Chapter 9]) allows to "recover" Y from the symmetric monoidal DG category QCoh(Y). On the other hand, the DG algebra H * (Y, O Y ) does not recover Y, unless Y is an affine DG scheme.
We characterize the tempered part of the automorphic Langlands category $$\mathfrak {D}({\text {Bun}}_G)$$ D ( Bun G ) using the geometry of the big cell in the affine Grassmannian. We deduce that, for G non-abelian, tempered D-modules have no de Rham cohomology with compact support. The latter fact boils down to a concrete statement, which we prove using the Ran space and some explicit t-structure estimates: for G non-abelian and $$\Sigma $$ Σ a smooth affine curve, the Borel–Moore homology of the indscheme $${\text {Maps}}(\Sigma ,G)$$ Maps ( Σ , G ) vanishes.
We consider the spherical DG category Sph G attached to an affine algebraic group G. By definition, Sph G := IndCoh(LSG(S 2 )) consists of ind-coherent sheaves on the (derived) stack of G-local systems on the 2-sphere S 2 . The three-dimensional version of the pair of pants endows Sph G with an E3-monoidal structure. More generally, for an algebraic stack Y and n −1, wewhere IndCoh0 is the sheaf theory introduced by Arinkin and Gaitsgory. The case of Sph G is recovered by setting Y = BG and n = 2.The cobordism hypothesis associates to Sph(Y, n) an (n + 1)-dimensional TFT, whose value on a manifold M d of dimension d n + 1 (possibly with boundary) is given by the topological chiral homology M d Sph(Y, n). In this paper, we compute such chiral homology, obtaining the Stokes style formulawhere the formal completion is constructed using the obvious projectionThe most interesting instance of this formula is for Sph G Sph(BG, 2), the original spherical category, and X a Riemann surface. In this case, we obtain a monoidal equivalenceis the stack of G-local systems on the topological space underlying X and H is a sheaf theory related to Hochschild cochains.
We strengthen the gluing theorem occurring on the spectral side of the geometric Langlands conjecture. While the latter embeds $IndCoh_N(LS_G)$ into a category glued out of 'Fourier coefficients' parametrized by standard parabolics, our refinement explicitly identifies the essential image of such embedding.
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